Bism Illah wa as salaamu alaykum wa Rahmat Allah wa Barakatuh.
There are various ways to study Arabic, Quran and Islam here in Madinah. The most famous is the Islamic University of Madinah, which takes applications from all over the world. Although there was much rumour that it only accepts students under 25 years of age, we can now say that is not true. Construction is in progress for a brand new Women's Campus at the university, so sisters will soon be able to apply as well - whoopee!
In addition, for men who work, there are free evening classes at Islamic University for Arabic. They use the Madinah books and do all four levels.
For women, there are two main programmes for Arabic language/Quran/Islamic Studies. One place is Mount Uhud School, which takes only married women. They teach at a fast pace and the course of studies is wide, varied, and intense. They teach from Lughat Al Arabia Bayna Yadayk Books and complete one book per semester. That means two books per year. On top of that, there are fiqh, aqeedah, hadith, and Quran memorization classes. There is a great deal of memorization of hadith and Islamic information involved and the sisters I have spoken to that attend or attended say it is difficult, mashaa'Allah. One sister said it is excellent if you already know Arabic reasonably well.
The other place, that I am attending, is Dar al Haafidhaat al Thania. It is really a tahfiz school for Quran memorization, but for years they have been teaching Arabic as well, mainly for the wives of the Jamiah students. They have Nursery, Pre-K, KG1, and KG2 classes for younger children, a special class for elementary school age girls which teaches Quran, tajweed, a bit of Arabic, and some Islamic studies, and the women's classes. There are levels and an ordinary tahfiz programme as well. The tamheedi level teaches from the book Lughat al Arabiya Bayna Yadayk. This is for those who need to get speaking the language. They also teach some basic tajweed and, of course, Quran memorization. The first level teaches Madinah books - level 1, the second level teaches Madinah books - level 2, the third level teaches Madinah books - level 3, and the fourth level teaches the second half of Madinah books - level 3.
Everywhere, in every area, neighbourhood, or district you will find tahfiz programmes. They run in the mornings and the afternoons and are free for all. If you want to memorize Quran, every facility is available for you, mashaa'Allah.
Alternatives for learning are to learn Arabic and then study at the foot of a shaykh. Several Jamiah students have said this is a much more effective and efficient way to study. There are classes held in Masjid an Nabawi, Masjid Qiblatain, and various other locations. Dr. Muhammad Al Jibaly holds classes for teenage boys once a week. Wives of shaykhs and qualified sisters hold classes for women, also.
This is a place where it is difficult not to gain 'ilm. Rather you bump into it even when you aren't trying. Just imagine, my husband called me at Asr time to tell me that there was a lesson on Quran recitation, the rules of tajweed, being given at the little masjid by our apartment. It was on the loudspeaker, so I could even open the window and hear it, mashaa' Allah. Truly, there is no excuse not to learn and grow.
Even though my husband is teaching here, he has gained immense knowledge just from his talks with his friends who attend the Jamiah or used to attend. These are brothers who have degrees in Shariah, Hadith, Fiqh, Aqeedah, and Da'wah. Some have Masters and some are working on their PhD's mashaa'Allah. May Allah give them tawfeeq and may we, who are so blessed to be invited to the home of our beloved Rasool, salla Allahu alahyi wa sallam, gain all the benefit Allah is offering us from this - ameen.